It is well known that the self-diffusion coefficient of a molecule is related in some way to its size. For a hard sphere in a fluid with viscosity ηs, the relationship is given by the Einstein-Stokes equation,
                    D        =                                            k              B                        ⁢            T                                6            ⁢            π            ⁢                                                  ⁢                          η              s                        ⁢            r                                              (        1        )            where r is the radius of the sphere and ηs is the viscosity of the solvent. This equation suggests that in a mixture with molecules of different radii, the diffusion coefficient Di of the ith component is
                              D          i                =                                            k              B                        ⁢            T                                6            ⁢            π            ⁢                                                  ⁢                          η              s                        ⁢                          r              i                                                          (        2        )            where ri is the radius of the ith component. From this, it can be concluded that the ratio of the diffusion coefficients of any two components in the mixture will depend only on the ratios of the sizes of the two molecules and is independent of any other properties of the fluid, such as its viscosity or temperature. Alternatively, Equation (2) implies that for a particular mixture, Diri is constant for all components in the mixture. In addition, Equation (2) states that there is a fixed relationship between the diffusion coefficients and the viscosity of the fluid (D∝1/ηs).
An application of the hard sphere model to oils may be found in Freedman et al. “A New NMR Method of Fluid Characterization in Reservoir Rocks: Experimental Confirmation and Simulation Results,” paper SPE 63214 presented at the 2000 SPE Annual Technical Conference and Exhibition, Dallas, 1-4 October; Lo et al. “Relaxation Time and Diffusion Measurements of Methane and Decane Mixtures,” The Log Analyst (November-December 1998); and Lo et al. “Correlations of NMR Relaxation Time with Viscosity, Diffusivity, and Gas/Oil Ratio of Methane/Hydrocarbon Mixtures,” in Proceedings of the 2000 Annual Technical Conference and Exhibition, Society of Petroleum Engineers (October 2000). These articles are hereby incorporated by reference herein in their entireties.
As shown in FIG. 1, diffusion coefficient D is related to the size of the molecule. Diffusion depends on the constituents (components and relative amounts) of the mixture under analysis. However, existing models do not adequately describe the physics of the system sufficient to identify the components of the mixture to an acceptable degree of certainty.
However, the hard sphere model is not adequate for describing more complicated molecules, such as oils, which are floppy chains. One example of the failure of the hard sphere model is evidenced by the measurements of diffusion and viscosity in alkanes and oils. In plots of log D versus log kT/η (see FIG. 2), the data all lie on a single line, regardless of the molecule's radii. These plots are in disagreement with Equation (2), which would imply that the slope depends on the radius of the molecule.
As will be shown below, elevated temperature and pressure may influence the modeling of complicated oils. Applications of the hard sphere model to oils do not adequately account for the effects of temperature and pressure on the diffusion coefficients and relaxation times.
Accordingly, it is one object of the present invention to provide a method to more appropriately model oils and oil mixtures.
It is another object of the present invention to provide a model that accounts for the temperature and pressure dependence of the diffusion coefficient and relaxation times over a wide range of temperatures and pressures.